Question: Mr. Garcia notes that in $84\%$ of the papers his former students have turned in, they cited their sources incorrectly. He plans to go through these papers at random to find one with incorrect citations for a lesson on citing with his current students. Assume that incorrect citations are independent between papers. Let $N$ be the number of papers Mr. Garcia selects until he finds one with the sources cited incorrectly. Find the probability that the $3^{\text{rd}}$ paper Mr. Garcia selects will be the first with the sources cited incorrectly. You may round your answer to the nearest hundredth. $P(N=3)=$
Solution: Without a fancy calculator For each paper's citations: $P({\text{incorrect}})=0.84$ $P(\text{correct}})=0.16$ If the $3^{\text{rd}}$ paper Mr. Garcia selects will be the first with the sources cited incorrectly, his sequence of paper citations must be "correct, correct, incorrect." It is reasonable to assume that Mr. Garcia has had more than $30$ papers turned in by former students. Since we are sampling less than $10\%$ of the population of papers, we can assume independence. $\begin{aligned} P(N=3)&=P(\text{CC}}{\text{I}}) \\\\ &=(0.16})(0.16})({0.84}) \\\\ &=(0.16)^2(0.84) \\\\ &=0.021504 \end{aligned}$ $P(N=3) \approx 0.022$